How many years could it take me to study and understand all the mathematics fields that exist so far?

My answer to a question on Quora: How many years could it take me to study and understand all the mathematics fields that exist so far?

If you mean understanding at the level of ability to do research work in every field of mathematics, then, I am afraid, there is no hope to achieve this goal. Mathematics expands, and the cutting edge of mathematical research moves further and further away from any fixed reference point, say, undergraduate mathematics. From the point of view of an aspiring PhD student, mathematics looks like New York in the Capek Brothers’ book A Long Cat Tale:

And New York – well, houses there are so tall that they can’t even finish building them. Before the bricklayers and tilers climb up them on their ladders, it is noon, so they eat their lunches and start climbing down again to be in their beds by bedtime. And so it goes on day after day.

It was written in the first half of the 20th century, but Joseph and Karel Capek understood thing or two about futurology (although the term “futurology”, most likely, did not exist in their time: they were the people who coined the word “robot”. We live in the world where, in almost every field of human endeavor, no-one can understand everything. The human civilization that we transform and build is immensely complex, and mathematics is perhaps its most complex part.

[For this post, I cannibalized some bits of my paper Mathematics for makers and mathematics for users; it discusses some relevant themes.]


Why is school 8 hours long?

My answer to a question on Quora: Why is school 8 hours long?

It was in primordial era, but, in my country, at my time at junior school (7 to 11 years old), school day was 4 lessons of 45 minutes long, with two breaks of 10 minutes and one break of 25 minutes in between, from 8:30 to 12:30 in the morning. There was some homework, but not very taxing. A plenty of time was free for whatever children wished to occupy themselves with. Parents were at work until 17:00.

A short school day is actually a physiological norm. Why in the UK, say, school day is abnormally long? Because it is an offence to leave children alone; the law is vague — see The law on leaving your child on their own, but it applies with unnecessary, in my opinion, rigour. Schools are forced to act as storage rooms for children while parents are at work.

Of course, in old times there were risks involved; legs and arms were broken while skiing (unsupervised), or playing ice hockey (also unsupervised), etc., but all that was seen as unavoidable and normal risks; there were no modern culture of over-protection which would, of course, cut accidents — but at expense of loss of child’s precious independence. Analysing now my and my friends’ behaviour of that time, I see that we were quite risk aware and knew how to avoid danger — it was a normal part of growing up.


Why don’t non-square matrices have determinants? The determinant is just the matrix’s scale factor (i.e. the “size” of the linear transformation), and I don’t see why a rectangular matrix wouldn’t have one.

My answer to a question in Quora: Why don’t non-square matrices have determinants? The determinant is just the matrix’s scale factor (i.e. the “size” of the linear transformation), and I don’t see why a rectangular matrix wouldn’t have one.

What follows is an answer that I would give to my students, if they asked it in the lecture. When you wish to generalise determinants to non-square matrices, but preserve their interpretation as “scale factors”, you have to preserve the multiplicativity of determinants: scale factors of consecutively executed transformations should multiply — otherwise why call them scale factors? Hence you perhaps wish to have, for this “extended” determinant, the property

\(\det AB = \det A \cdot \det B\) whenever the product \(AB\) exists.

Perhaps you would also wish this “new extended” determinant to coincide with the traditional determinant when applied to square matrices.

Alas, this is impossible: take

\(A = \begin{pmatrix} 1 & 1 \end{pmatrix} \)


\(B = \begin{pmatrix} 1 \\ 1 \end{pmatrix} \)


\(AB = \begin{pmatrix} 2\end{pmatrix} \)


\(BA = \begin{pmatrix} 1 &1 \\ 1&1 \end{pmatrix} \)


\(\det A \cdot \det B = \det AB = \det(2) = 2 \)


\(\det B \cdot \det A = \det BA = \det\begin{pmatrix} 1 &1 \\ 1&1 \end{pmatrix} = 0, \)


\(\det A \cdot \det B \ne \det B \cdot \det A\)

—an obvious contradiction.


What is the most common reaction when you tell people you’re a mathematician/you study mathematics?

My answer to a question in Quora: What is the most common reaction when you tell people you’re a mathematician/you study mathematics?

It was quite a while when I last heard any offensive comment. But I am bald, fat, wear large round glasses and a tweed jacket with a tie. I look every bit as a stereotypical university lecturer of certain age, who I actually am. In a conversation of that kind, I am quickly left in peace, which suits me. Why I am left in peace? Because I treat people who talk to me with respect and kindness — but without showings signs of weakness. These are, basically, professional traits of a teacher, no more than that.

Why do some strangers ask questions and make comments of the kind quoted in previous answers in this thread? Because they take at face value a false stereotype: that mathematicians are emotionally vulnerable. Normally, they are not – this is in the nature of mathematics. After all, when a mathematician proves a theorem, she or he is the only person in the entire world who knows The Truth. Well, it could be a small theorem and a small truth, but it is being alone on a intellectual mountain peak that matters.

Even if you have just started to learn mathematics, recognise mathematics as a weapon of personal intellectual empowerment – and use it. Notice, and pay attention to, this remarkable new feeling that gradually develops in your soul: that you know certain things with absolute certainty, and that you can prove to others that your understanding is correct.

Perhaps I have to add that after 40+ years spent at a blackboard in front of audiences of up to 400 students, I, most likely, do not look as a vulnerable person. If you feel offended by remarks that strangers might make about mathematics and mathematicians, try this simple remedy: get seriously involved in teaching of mathematics — and everything in the world around you will return to its proper place.


What is the counterargument to the statement “theoretical physics is harder than pure mathematics because you need to know the math and the physics to be a physicist but only math to be a mathematician”?

My answer to a question in Quora: What is the counterargument to the statement “theoretical physics is harder than pure mathematics because you need to know the math and the physics to be a physicist but only math to be a mathematician”?

My world outlook is very skewed, I know mostly mathematicians.

While writing this answer, I went in my head through the list of mathematicians who I met personally, and who were known for contribution to development of other fields, first of all, physics, but also, say, genetics and molecular biology, or who were able to use physics as a source of ideas and problems for their mathematical research. They all are, first and foremost, excellent hardcore mathematicians.

It is my conjecture that work at a serious level in both serious pure mathematics and theoretical physics require capacity for a specific kind of abstract thinking: ability to keep in mind a mental image of the world hierarchically built from increasing levels of abstraction, and ability to live in this world and freely move from one level to another. This could be the Platonic world of mathematics or the world of relativistic quantum physics — but I dare to suggest that the two strands of thinking, mathematical and physical, start to converge at higher level of development.

I do not claim that these my observations have any depth; they are simply an output of five minutes of silent contemplation.

Speaking about mathematicians and physicists, what really astounds me is the description of the work of a theoretical physicist in Vasily Grossman’s novel Life and Fate(the name of the character is Victor Schtrum; it is not clear who was his real life prototype). How could Grossman know? Strongly recommend.


Ofsted Survey – Help Please!

Reposted from Let Our Kids Be Kids: Ofsted Survey – Help Please!

Sometimes it can feel like you are trying to turn an oil tanker when you are fighting for change – the cogs in the machines of power can take an awfully long time to turn!  There are many oil tankers in the education system… all stuck in their seas of thick mud!  We have been trying to push in as many ways as we can think of to get back on track towards a system where children are treated as children, teachers are free to teach a joyous curriculum and high stakes tests are no longer the foundation of our school accountability system.

Ofsted are one of the biggest and most powerful tankers in the education sea.  And yet perhaps there is change on it’s way?!  Currently Ofsted are starting to question how schools are inspected…

Perhaps data from tests shouldn’t be the only tool used to measure schools by? Could child well-being be of equal importance in the success of a school?  (Who’d have thought?!)  Maybe a school’s overall approach to a broad and balanced curriculum could be more important than just how well they can cram children through SAT tests? (Again… who knew?!)

Ofsted are currently conducting a survey to answer these questions and we want to formulate our own response from parents.  It’s fantastic that these questions are being asked but we know that this oil tanker can’t move all by itself – it needs a MASSIVE push of parent, teacher and professional power to help it on it’s way.


We can’t do this without you!! Thank you!


What made you go from hating math to loving it?

My answer to a question on Quora: What made you go from hating math to loving it?

Nothing. I always loved mathematics, and luckily, nothing ruined my love. A much more detailed answer can be found in my paper

A. Borovik, Comments on “Stop Ruining Math! Reasons and Remedies for the
Maladies of Mathematics Education” by Rachel Steinig
, The De Morgan Gazette 8 no. 2 (2016) 9-18. bit.ly/2b8nSht

Rachel Steinig was a 16 years old student who wrote about school mathematics education from a position of a student. For many months she has been surveying people around her about how math was ruined for them. She said in her paper (R. M. Steinig, Stop Ruining Math! Reasons and Remedies for the Maladies of Mathematics Education. J. Humanistic Mathematics, 6 no. 2 (July 2016), 128–147.):

I’ve asked my friends and relatives, posted on Facebook, and asked teachers and parents. Even though everyone’s story was different, there were some common themes running throughout. [. . . ] Just to be clear, these are not my opinions of what ruins math—these are the results of surveying many people of all ages, education levels, and attitudes. [pp. 128–129]

Rachel Steinig’s observations were fresh and incisive. On reading them, I have been struck by realisation that I became a professional mathematician only because, in my school years, almost in every situation described by Rachel Steinig, I had experiences exactly opposite to the ones reported by her. In short, I became a mathematician only because

nothing had ruined my mathematics.


Chalk up your opposition to tests!

Reposted from Chalk up your opposition to tests!

Over the next couple of months parents, teachers, nursery leaders, child development experts and of course, children, are chalking up their opposition to high stakes testing in primary school!  It’s easy to get involved…

Grab some chalk and write the words #PlayNotTestsAt4 on a pavement near you – then post a pic on social media using #PlayNotTestsAt4 and linking to @morethanascore who are leading the campaign.

The more people who get involved in this creative protest the better – to show the government that we understand the importance of PLAY in schools and the nonsense of TESTING 4 YEAR OLDS!  We can’t believe that in 2019, when we know so much about child development and the growing young brain that we are having to defend children’s right to PLAY… yet this is the case as more and more academic style learning comes down the primary school curriculum and into Key Stage 1 and Reception.

It’s our firm belief that children under the age of 7 should be learning through PLAY most of the time – directed by adults yes to ensure safety and maximise learning but PLAY should be the focus of the child’s day at school to encourage an early love of school and a relaxed, individualised path towards more focused learning in the later stages of the curriculum.

Scandinavian countries use this approach, recognising that children are so widely different developmentally until they are 7 that it’s pointless and damaging to try and ‘standardise’ their learning before this age… this introduction of tests for 4 year old in English and maths, in their first few weeks at school, in demonstrative of this government’s complete lack of understanding of child development and cognitive learning  in childhood. #PlayNotTestsAt4 is such a simple message but an absolutely vital one to protect PLAY for our children!

For more details and ways to get involved please visit More Than A Score!


Is there any idempotent matrix that is not normal?

My answer to a Question on Quora: Is there any idempotent matrix that is not normal?

I came up with exactly the same answer

\(v_1 = \left[\begin{array}{cr} 1 & -1 \\0 & 0 \end{array}\right] \)

as Alex Eustis did, by looking at perhaps one of the simplest possible examples: a \(2\times 2\) matrix diagonalisable with eigenvalues 1 and 0 in the basis made of vectors

\(v_1 = \left[\begin{array}{c} 1 \\0 \end{array}\right] \; \mbox{ and }\; v_2= \left[\begin{array}{c} 1 \\1 \end{array}\right]\)

which are not orthogonal to each other. Can you suggest a simpler non-orthogonal basis in \(\mathbb{R}^2\)?


James D. Watson: “Extend yourself intellectually through courses that initially frighten you”

The famous geneticist James Watson, of the double helix fame, about his relations with mathematics:

All through my undergraduate days I worried that my limited mathematical talents might keep me from being more than a naturalist.  In deciding to go for the gene, whose essence was surely in its molecular properties, there seemed no choice but to tackle my weakness head-on.  Not only was math at the heart of virtually all physics, but the forces at work in three-dimensional molecular structures could not be described except with math.  Only by taking higher math courses would I develop sufficient comfort to work at the leading edge of my field, even if I never got near the leading edge of math.  And so my Bs in two genuinely tough math courses were worth far more in confidence capital than any A I would likely have received in a biology course, no matter how demanding.  Though I would never use the full extent of the analytical methods I had learned, the Poisson distribution analyses needed to do most phage experiments soon became satisfying instead of a source of crippling anxiety.

[From J. D. Watson, Avoid Boring People, Vintage Books, New York, 2010, p. 51]